Theorem iunsuc 4467

Description: Inductive definition for the indexed union at a successor. (Contributed by Mario Carneiro, 4-Feb-2013.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)

Hypotheses

Ref	Expression
iunsuc.1	|- A e. _V
iunsuc.2	|- ( x = A -> B = C )

Assertion

Ref	Expression
iunsuc	|- U_ x e. suc A B = ( U_ x e. A B u. C )

Distinct variable groups:   x, A   x, C

Allowed substitution hint:   B( x)

Proof of Theorem iunsuc

Step	Hyp	Ref	Expression
1		df-suc 4418	. . 3 |- suc A = ( A u. { A } )
2		iuneq1 3940	. . 3 |- ( suc A = ( A u. { A } ) -> U_ x e. suc A B = U_ x e. ( A u. { A } ) B )
3	1, 2	ax-mp 5	. 2 |- U_ x e. suc A B = U_ x e. ( A u. { A } ) B
4		iunxun 4007	. 2 |- U_ x e. ( A u. { A } ) B = ( U_ x e. A B u. U_ x e. { A } B )
5		iunsuc.1	. . . 4 |- A e. _V
6		iunsuc.2	. . . 4 |- ( x = A -> B = C )
7	5, 6	iunxsn 4004	. . 3 |- U_ x e. { A } B = C
8	7	uneq2i 3324	. 2 |- ( U_ x e. A B u. U_ x e. { A } B ) = ( U_ x e. A B u. C )
9	3, 4, 8	3eqtri 2230	1 |- U_ x e. suc A B = ( U_ x e. A B u. C )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   _Vcvv 2772   u. cun 3164   {csn 3633   U_ciun 3927   suc csuc 4412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-iun 3929  df-suc 4418

This theorem is referenced by: (None)